Null controllability of the parabolic spherical Grushin equation - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue ESAIM: Control, Optimisation and Calculus of Variations Année : 2022

Null controllability of the parabolic spherical Grushin equation

Cyprien Tamekue

Résumé

We investigate the null controllability property of the parabolic equation associated with the Grushin operator defined by the canonical almost-Riemannian structure on the 2-dimensional sphere S 2. This is the natural generalization of the Grushin operator G = ∂ 2 x + x 2 ∂ 2 y on R 2 to this curved setting, and presents a degeneracy at the equator of S 2. We prove that the null controllability is verified in large time when the control acts as a source term distributed on a subset ω = {(x 1 , x 2 , x 3) ∈ S 2 | α < |x 3 | < β} for some 0 ≤ α < β ≤ 1. More precisely, we show the existence of a positive time T * > 0 such that the system is null controllable from ω in any time T ≥ T * , and that the minimal time of control from ω satisfies T min ≥ log(1/ √ 1 − α 2). Here, the lower bound corresponds to the Agmon distance of ω from the equator. These results are obtained by proving a suitable Carleman estimate by using unitary transformations and Hardy-Poincaré type inequalities to show the positive nullcontrollability result. The negative statement is proved by exploiting an appropriate family of spherical harmonics, which concentrates at the equator, to falsify the uniform observability inequality.
Fichier principal
Vignette du fichier
tamekue2021.pdf (802.41 Ko) Télécharger le fichier
Origine : Fichiers éditeurs autorisés sur une archive ouverte

Dates et versions

hal-03719160 , version 1 (11-07-2022)
hal-03719160 , version 2 (12-12-2022)

Identifiants

Citer

Cyprien Tamekue. Null controllability of the parabolic spherical Grushin equation. ESAIM: Control, Optimisation and Calculus of Variations, 2022, 28, pp.70. ⟨10.1051/cocv/2022055⟩. ⟨hal-03719160v2⟩
78 Consultations
36 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More